Editing Ellipse (section)

=== Definition of conjugate diameters ===
[[File:Parallelproj-kreis-ellipse.svg|400px|thumb|Orthogonal diameters of a circle with a square of tangents, midpoints of parallel chords and an affine image, which is an ellipse with conjugate diameters, a parallelogram of tangents and midpoints of chords.]]
{{Main|Conjugate diameters}}

A circle has the following property:
: The midpoints of parallel chords lie on a diameter.

An affine transformation preserves parallelism and midpoints of line segments, so this property is true for any ellipse. (Note that the parallel chords and the diameter are no longer orthogonal.)

; Definition:
Two diameters <math>d_1,\, d_2</math> of an ellipse are ''conjugate'' if the midpoints of chords parallel to <math>d_1</math> lie on <math>d_2\ .</math>

From the diagram one finds:
: Two diameters <math>\overline{P_1 Q_1},\, \overline{P_2 Q_2}</math> of an ellipse are conjugate whenever the tangents at <math>P_1</math> and <math>Q_1</math> are parallel to <math>\overline{P_2 Q_2}</math>.

Conjugate diameters in an ellipse generalize orthogonal diameters in a circle.

In the parametric equation for a general ellipse given above,
<math display="block">\vec x = \vec p(t) = \vec f\!_0 +\vec f\!_1 \cos t + \vec f\!_2 \sin t,</math>

any pair of points <math>\vec p(t),\ \vec p(t + \pi)</math> belong to a diameter, and the pair <math>\vec p\left(t + \tfrac{\pi}{2}\right),\ \vec p\left(t - \tfrac{\pi}{2}\right)</math> belong to its conjugate diameter.

For the common parametric representation <math>(a\cos t,b\sin t)</math> of the ellipse with equation <math>\tfrac{x^2}{a^2}+\tfrac{y^2}{b^2}=1</math> one gets: The points 
:<math>(x_1,y_1)=(\pm a\cos t,\pm b\sin t)\quad </math> (signs: (+,+) or (−,−) )
:<math>(x_2,y_2)=({\color{red}{\mp}} a\sin t,\pm b\cos t)\quad </math> (signs: (−,+) or (+,−) )
:are conjugate and 
:<math>\frac{x_1x_2}{a^2}+\frac{y_1y_2}{b^2}=0\ .</math>
In case of a circle the last equation collapses to <math>x_1x_2+y_1y_2=0\ . </math>